
where the integer \(n\)  labels different coherence orders. The SP effect has been extensively studied using macroscopic structures such as gratings and particle arrays [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, as well as atomic layers in stacked van der Waals materials [11]}. In addition, electrons act in unison if they are bunched within small beam regions compared to the emission wavelength in the moving frame, giving rise to superradiant light generation ranging from the terahertz [12]} to the x-ray [13]}, [14]} spectral domains. Incidentally, the inverse process (electron energy gain near an illuminated grating) has also been demonstrated [15]} and explored as a route to realise laser-based table-top particle accelerators [16]}.
